Author(s): ChunGil Park
It is shown that every almost linear mapping h : A → B of a unital Lie JC ∗ -algebra A to a unital Lie JC ∗ -algebra B is a Lie JC ∗ - algebra homomorphism when h (2 n u ◦ y ) = h (2 n u ) ◦ h ( y ), h (3 n u ◦ y ) = h (3 n u ) ◦ h ( y ) or h ( q n u ◦ y ) = h ( q n u ) ◦ h ( y ) for all y ∈ A , all unitary elements u ∈ A and n = 0 , 1 , 2 , ··· , and that every almost linear almost multiplicative mapping h : A → B is a Lie JC ∗ -algebra homomorphism when h (2 x ) = 2 h ( x ), h (3 x ) = 3 h ( x ) or h ( qx ) qh ( x ) for all x ∈ A . Here the numbers 2 , 3 ,q depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings. Moreover, we prove the Cauchy–Rassias stability of Lie JC ∗ -algebra homomorphisms in Lie JC ∗ - algebras, and of Lie JC ∗ -algebra derivations in Lie JC ∗ -algebras. Mathematics Subject Classification: 17B40, 39B52, 46L05, 17A36. Keywords and Phrases: Lie JC ∗ -algebra homomorphism. Lie JC ∗ -algebra derivation, stability, linear functional equation.